Function increment is a mathematical concept, which refers to the change of output value when the input value of a function changes. It can help us to calculate the variation of the function, so as to better understand the characteristics of the function.

Brief introduction to nouns:

A continuous function refers to the function y=f(x). When the change of independent variable X is small, the change of dependent variable Y is also small. For example, the temperature changes with time, as long as the time change is small, the temperature change is small; For another example, the displacement of a free-falling body changes with time. As long as the time change is short enough, the displacement change is also small. For this phenomenon, the dependent variable changes continuously around the independent variable.

Rules:

Theorem 1 A finite function that is continuous at a certain point is still a continuous function at that point after a finite number of operations of sum, difference, product and quotient (denominator is not 0). Theorem 2 The inverse function of a continuous monotone increasing (decreasing) function is also continuously monotone increasing (decreasing). Theorem 3 The composite function of a continuous function is continuous. These properties can be derived from the definition of continuity and the related properties of limit.

Maximum value:

A continuous function on a closed interval must be able to obtain the maximum and minimum values in this interval. The so-called maximum value means that there is a point x0 on [a, b], so that for any x∈[a, b], there is f(x)≤f(x0), then f(x0) is called the maximum value of f(x) on [a, b]. The minimum value can also be defined in the same way, but the sign of the inequality above is reversed.

Inverse function continuity:

If the function f is strictly monotonically continuous in its domain d, then its inverse function f- 1 is also strictly monotonically continuous in its domain f(D) (that is, the value domain of f). It is proved that a strictly monotone function must have a strictly monotone inverse function, and the proof of monotonicity is the same as that of the inverse function, so it is only necessary to prove that the inverse function is also continuous in its domain. Let f be a strictly simple increasing function defined on d.